Puzzle Types
Puzzles in competitive puzzling are language neutral (so they also don’t require trivia knowledge) logic puzzles. They are generally written to have a logical solving path and only one solution. There are many many types, though they tend to fall into various categories. GM Puzzles uses the categorisation of Loop/Path puzzles, Number Placement, Object Placement, Region Division and Shading, but sometimes puzzles will straddle these genres.
It is natural to learn new types as you go along, rather than trying to master everything before starting to compete. At the bottom is a library of some standard puzzle types from the UK Puzzle League, with instructions, examples and some practice puzzles if you need help with the logic, otherwise there are many blogs and the below sources of puzzles.

Grandmaster Puzzles (or GM Puzzles) has a range of PDF based puzzles which give a difficulty rating and have some excellent entry-level puzzles. They also sell larger PDF ebooks of puzzles.

Tambox sell an excellent e-magazine, Pulze, with a range of puzzles, as well as their weekly Puzzler’s Bulletin.

Logic Masters India have their Puzzle Expo, which has many puzzles for online solving.

Logic Masters Germany have their Puzzle Portal with a wide range of puzzles, with online solving options.

Puzz Link is not the most beautiful website, but it does have thousands of puzzles and is easy to search to find lots of examples to solve, often with online solving options.

Puzzle Duel has a daily puzzle with live online leaderboards as well as a good archive of past puzzles. They also focus on one specific puzzle a couple of times each year and have a whole month of bonus puzzles. It has its own online solving platform.
UK Puzzle League Archive